Multiple choice

If $2 + 3i$ is one of the roots of the equation $2x^3 - 9x^2 + kx - 13 = 0,\,\, k \in R$, then the real root of this equation :

  1. exist and is equal to $\displaystyle \frac{1}{2}$

  2. does not exist

  3. exist and is equal to 1
  4. exist and is equal to $- \displaystyle \frac{1}{2}$
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A Correct answer
Explanation

Since coefficients are real, if 2 + 3i is a root, 2 - 3i must also be a root. The sum of roots is (2 + 3i) + (2 - 3i) + r = 9/2, so 4 + r = 4.5, which means r = 0.5 or 1/2.

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